Polynomial tau-functions of BKP and DKP hierarchies

Abstract

In a previous paper we constructed all polynomial tau-functions of the 1-component KP hierarchy, namely, we showed that any such tau-function is obtained from a Schur polynomial s λ (t) by certain shifts of arguments. In the present paper we give a simpler proof of this result, using the (1-component) boson-fermion correspondence. Moreover, we show that this approach can be applied to the s-component KP hierarchy, using the s-component bosonfermion correspondence, finding thereby all its polynomial tau-functions. We also find all polynomial tau-functions for the reduction of the s-component KP hierarchy, associated to any partition consisting of s positive parts.

Polynomial tau-functions of the KP, BKP, and the s-component KP hierarchies

“…for any choice of constant vectors c j = (c j,k ) k∈N . This recovers the description of all polynomial tau-functions of the BKP hierarchy of [9] and, in particular, the result of You [16] that Schur Q-polynomials are polynomial tau-functions of the BKP hierarchy.…”

“…ϕ k l (1), it is a polynomial tau-function. d) Following [9] Proposition 3, any polynomial tau-function has the form…”

Intersection numbers on $$ {\overline{M}}_{g,n} $$ and BKP hierarchy

“…For example, Schur Q-functions provide all possible polynomial solutions of the BKP hierarchy [11,20,30,39].…”

“…From Theorem 1 it follows that after rescaling of times, τ BGW (t/2, N ), these functions are also solutions of the BKP hierarchy. Hence [20] they are shifted Schur Q-functions.…”

BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions

“…See e.g. [19,22,31,36,37,41] for more information about the BKP hierarchy, and see the book [6] for an introduction of affine coordinates of a BKP tau-function. Let τ = τ (t) be a tau-function of the BKP hierarchy with τ (0) = 1 where t = (t 1 , t 3 , t 5 , • • • ) are the time variables, then τ can be represented as a Bogoliubov transform in the fermionic Fock space:…”